Multilinear formulations for computing Nash equilibrium of multi-player matrix games

TL;DR

This paper introduces multilinear and mixed-integer multilinear programming approaches to compute Nash equilibria in multi-player games, demonstrating that the multilinear feasibility program outperforms existing algorithms in speed.

Contribution

It presents novel multilinear and mixed-integer formulations for multi-player Nash equilibrium computation, with the multilinear feasibility program showing superior performance.

Findings

Multilinear feasibility program finds Nash equilibria faster than existing methods.

Mixed-integer formulations do not outperform current algorithms for multi-player games.

The approach offers a competitive alternative to traditional algorithms, especially for large games.

Abstract

We present multilinear and mixed-integer multilinear programs to find a Nash equilibrium in multi-player noncooperative games. We compare the formulations to common algorithms in Gambit, and conclude that a multilinear feasibility program finds a Nash equilibrium faster than any of the methods we compare it to, including the quantal response equilibrium method, which is recommended for large games. Hence, the multilinear feasibility program is an alternative method to find a Nash equilibrium in multi-player games, and outperforms many common algorithms. The mixed-integer formulations are generalisations of known mixed-integer programs for two-player games, however unlike two-player games, these mixed-integer programs do not give better performance than existing algorithms.